## Dirac-Appell Sequences

The Pincherle derivative $[T^n(L,R),R] = \frac {d}{dL}T^n(L,R)= n \cdot T^{n-1}(L,R)$ is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence.

The fundamental D-A sequence can be defined as the sequence $\delta^{(n)}(x)$ with $L = -x$ and $R = D = d/dx$ and e.g.f. $e^{tD} \delta(x) = \delta(x+t)$. Another example is provided by OEIS A099174 with the D-A sequence $H_n(x) = h_n(D) \delta(x)$ where $h_n(x)$ are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

A) $h_n(x) = e^{D^2/2} x^n$

B) $h_n(x) = R_h^n \; 1 = R_h \; h_{n-1}(x) = (x + D) \; h_{n-1}(x) = e^{D^2/2} \; x \; e^{-D^2/2} \; h_{n-1}(x)$

C) $L_h \; h_n(x) = D \; h_n(x) = n \; h_{n-1}(x)$

D) $\exp[h.(x)t]=e^{D^2/2} e^{xt}= e^{t^2/2} \; e^{xt}$.

The corresponding D-A sequence $H_n(x) = h_n(D) \delta(x)$ can be characterized similarly in various ways:

A2) $H_n(x) = e^{x^2/2} \; \delta^{n}(x) = e^{x^2/2} D^n \; \delta(x)$

B2) $H_n(x) = R_H^n \; \delta(x) = R_H \; H_{n-1}(x) = (D - x) \; H_{n-1}(x) = e^{x^2/2} \; D \; e^{-x^2/2} \; H_{n-1}(x)$

C2) $L_H \; H_n(x) = -x \; H_n(x) = n \; H_{n-1}(x)$

D2) $\exp[t H.(x)] = e^{x^2/2} \; e^{t D} \delta(x) = e^{x^2/2} \delta(x+t)= e^{t^2/2} \delta(x+t) = e^{t^2/2} \; e^{t D} \delta(x)$.

See the MSE-Q “Extending an identity for the Dirac delta function” for a presentation of

$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$,

which is useful for checking the identities.

The reps of the raising ops as conjugated versions of the iconic raising ops are easily  understood in terms of umbral  compositional  inversion

$p_n(\hat{p}.(x)) = x^n = \hat{p}_n(p.(x))$.

As discussed in the Bernoulli Appells post, one generator for the Appell umbral compositonal inverse sequence is $\frac{1}{e^{a.D}} \; x^n = \hat{p}_n(x)$. For our Hermite polynomials, the umbral compositional inverse sequence of polynomials is given by (cf. A066325)

$\hat{h}_n(x) = e^{-D^2/2} \; x^n = (x - D)^n \; 1 = (x-D) \; \hat{h}_{n-1}(x)$,

so

$e^{D^2/2} \; x \; e^{-D^2/2} \; h_n(x) = e^{D^2/2} \; x \; h_n(\hat{h}.(x)) = e^{D^2/2} \; x^{n+1} = h_{n+1}(x)$.

The basic Appell power series terms $x^n$ with raising op $x$ may be replaced by the basic D-A series terms $\delta^{(n)}(x)$ with raising op $D$ and the same machinations hold.

Multiplying an Appell polynomial  by another corresponds to convolving the corresponding D-A functions, which have representations as Cauchy complex-contour-integral operators and band-limited Fourier transforms, same as for the fractional calculus discussed in previous entries. For example, convolution of the basic D-A sequence gives

$\delta^{(m)}(x) \otimes \delta^{(n)}(x) = \int_0^{x} \delta^{(m)}(u)\; \delta^{(n)}(x-u)du$ $= \frac{\mathrm{d^n} }{\mathrm{d} x^n}\int_0^{x} \delta^{(m)}(u)\; \delta(x-u)du = \delta^{(m+n)}(x)$.

The relations in D2 can be expressed in other ways using basic properties of the Dirac delta function and identities in operator calculus:

$e^{t^2/2} \; \delta(x+t) = \delta[e^{-t^2/2}(x+t)] = e^{tD}(e^{-t^2/2})^{xD} \; \delta(x)= e^{tD}(e^{-t^2xD/2}) \; \delta(x) =e^{tD}(e^{-t^2\phi.(:xD:)/2}) \; \delta(x) = e^{tD} \exp[(e^{-t^2}-1):xD:] \; \delta(x)$

where by definition $(:xD:)^n = x^n D^n$ and $\phi_n(x)$ are the Bell / Touchard / exponential polynomials, discussed in several previous entries.

Inspecting the characterizations, we see that, given a  basic sequence of functions with raising and lowering ops $R_b \; b_n(x)=b_{n+1}(x)$ and $L_b \; b_n(x) = n \; b_{n-1}(x)$, a generalized Appell sequence $G_n(x)$ can be formed from a regular Appell sequence with $a_0 = 1$ and $g_n(x) = (a. + x)^n= e^{a.D}x^n$ with

A3) $G_n(x) = g(R_b) \; b_0(x) = (a. + R_b)^n \; b_0(x) = (a. + b.(x))^n = \sum_{k=0}^n \; \binom{n}{k} \; a_k \; b_{n-k}(x)$

$= e^{a.L_b} \; b_n(x)$

B3) $G_n(x) = R_G^n \; G_0(x) = R_G \; G_{n-1}(x) = e^{a.L_b} \; R_b \; \frac{1}{e^{a.L_b}} \; G_{n-1}(x)$

$=e^{a.L_b} \; R_b \; \frac{1}{e^{a.L_b}} \; e^{a.L_b} \; b_{n-1}(x) = e^{a.L_b} \; R_b \; b_{n-1}(x) = e^{a.L_b} \; b_n(x)$

C3) $L_b \; G_n(x) = L_b \; e^{a.L_b} \; b_n(x)= n \; e^{a.L_b} \; b_{n-1}(x)=n \; G_{n-1}(x) = L_G \; G_n(x)$

D3) $e^{G.(x)t} = e^{a.L_b} \; e^{b.(x)t} = e^{a.L_b} \; e^{t R_b} \; b_0(x)=e^{a.L_b} \; e^{t R_b} \; G_0(x)$.

So,

$L_G = L_b$

and

$R_G = e^{a.L_b} \; R_b \;\frac{1}{e^{a.L_b}} = R_b \; + \; [e^{a.L_b},R_b] \;\frac{1}{e^{a.L_b}} = R_b + \frac{d(e^{a.L_b})}{dL_b} \; \frac{1}{e^{a.L_b}}$

$= R_b + \frac{d(\ln[e^{a.L_b}])}{dL_b} = R_b - \frac{d(\ln[1/e^{a.L_b}])}{dL_b}$.

Be careful with the interpretation of the umbral operations here; in general,

$\ln[e^{a.L_b}] = \ln[\sum_{n \ge 0} \; \frac{a_n L_b^n}{n!}] \neq (a.L_b)$

and

$\frac{1}{e^{a.x}} \neq e^{-a.x}$

unless $(a.)^n = a_n = c^n$,

where $c$ is a non-umbral expresion independent of the value of $x$, e.g., $c = t^2/2$ for the modified Hermite polynomials, but it could also be some operator that doesn’t act on $x$.

An Appell sequence has the binomial shift property

$(g.(x) + y)^n = (a. + x +y)^n = g_n(x+y) = e^{yD} \; g_n(x)= e^{(a.+y)D} \; x^n$,

but the associated generalized Appell sequence will not possess this property unless the underlying base sequence has it, i.e., unless $(b.(x) + y)^n = b_n(x+y)$. Only then will

$G_n(x+y) = (a. + b.(x+y))^n = (y + a. + b.(x))^n = (y + G.(x))^n$.

In particular the Dirac-Appell sequence does not inherit this property since

$\delta^{(n)}(x+y) = e^{yD}\; \delta^{(n)}(x) \neq (\delta^{(.)}(x) + y)^n$.

Instead, the binomial convolution shift property for the basic Appell power basis translates into a multiplication by an exponential of the D-A sequence:

$e^{yL_b} b_n(x) = e^{-yx} \delta^{(n)}(x) = (y + \delta^{(.)}(x))^n = (y + b.(x))^n$,

which can be confirmed by using the basic identity above for products of the power monomials with the derivatives of the delta function or through the inverse Laplace transform acting on $(p+y)^n$ with a simple translation of the complex integration variable $p$.

Related Stuff:

One-dimensional Quasi-exactly Solvable  Schrodinger Equations” by Turbiner, p. 17